The present application relates to electronic-design-automation (EDA) tools, and more particularly to modeling of the physical properties of devices disposed in integrated circuits designed using such tools.
As transistors, such as field-effect or bipolar transistors, used to form an integrated circuit (IC) are scaled to ever smaller dimensions, the mathematical models used to describe the physical behavior of such transistors/devices need to be reconsidered. Field-effect transistors are widely used as switching elements in logic circuits. In a field-effect transistor in the on-state, charges flow from the source region to the drain region via a channel. In the off-state, charges are blocked from flowing between the source and drain regions. A gate electrode is used to turn on/off a conduction path (channel), thereby to control the flow of current between the source and drain regions.
Transistors are typically modeled either macroscopically using equations to model relationships between charge, potential, and current flux at contacts, or microscopically using a detailed representation of the device in one, two, or three spatial dimensions. The macroscopic approach, or “compact model” approach, has the advantage of computational efficiency but is relatively inefficient at predicting the effect of design changes on the device behavior. Microscopic models may be used to predict the impact of relatively minor differences on a device physical behavior as well as predicting the behaviors of substantially different devices formed in the same or similar materials.
In a typical microscopic model, a physical representation of the device is created by partitioning the device into regions, and partitioning the regions into elements bounded by vertices. Physics-based equations govern how fields vary between elements or vertices, and charge fluxes are modeled in terms of these fields. Charges are characterized using a “carrier” model, where quanta of charge is considered to be transported by individual quasi-particles called “electrons” (negatively charged) or “holes” (positively charged). The term quasi-particle describes a quantum of charge which exhibits particle-like behavior in a semiconducting crystal. This behavior includes characteristics such as velocity, momentum, energy, and mass. These quasi-particles or carriers travel between various regions of the device at speeds that depend on a number of factors, such as the electrostatic potential, the density of carriers, the lattice temperature, and the like.
A well-known model for characterizing the physics of solid state devices is the drift-diffusion model in accordance with which carrier velocity is considered dependent on local fields and their gradients. Drift-diffusion model, which is a local model, determines a series of fields and/or their gradients at any given point to predict the carrier velocity and/or flux at that point. A non-local model may additionally consider the value of fields at other points in the device.